When j=1, each of the series (28.23.6)–(28.23.13)
converges for all z. When j=2,3,4 the series in the even-numbered
equations converge for ℜ⁡z>0 and |cosh⁡z|>1, and the
series in the odd-numbered equations converge for ℜ⁡z>0 and
|sinh⁡z|>1.

For proofs and generalizations, see
Meixner and Schäfke (1954, §§2.62 and 2.64).